Chemistry Reference
In-Depth Information
σ
v
σ
v
C
2
C
2
σ
v
′
σ
v
′
Figure 2.1
The symmetry elements of water and fluorobenzene; in this case, each element
has only one corresponding operation.
C
2
axis and two vertical mirror planes,
σ
v
. These molecules do not have any other
rotation axes or mirror planes and do not have an inversion centre; so, according to the
operations met so far, this set completely describes the symmetry of both molecules. The
two molecules are said to belong to the same symmetry point group because they have
exactly the same set of valid symmetry operations.
σ
v
and
2.3 Closed Groups and New Operations
2.3.1 Products of Operations
In general, the idea of a point group is a mathematical abstraction that helps us classify
molecular geometry. A point group is a list of all symmetry operations that an object which
belongs to the group can undergo and remain apparently unchanged. The set of operations
that form a group must be complete, in the sense that if any two members of the group
are applied in succession the result must also be a single operation which is a member
of the group. This means that the group is 'closed', i.e. it is not possible to generate a
new symmetry operation by combining those in the group. This property of groups can be
useful: ensuring that the group of operations is closed is one way of checking that all the
operations that are possible have been identified. The requirement for a group to be closed
also leads to further symmetry operations, as will be shown in the following sections.
Multiplication Table for H
2
O
To check that a group is closed we form the product of each pair of operations within
the group and find the single equivalent operation that achieves the same end point. The
product of a pair of symmetry operations is defined as the result of applying them in
succession. Taking H
2
O as an example,
C
2
σ
v
is the product of a vertical reflection through
the molecular plane and a 180
◦
rotation and is achieved by carrying out the reflection
followed by the rotation. This would be one possible combination of operations for the
H
2
O molecule; and if the group is closed, all such products should be equivalent to a single
operation. A complication can arise in finding the single operation that is equivalent to the
product. For the example of
C
2
σ
v
the hydrogen atoms would be swapped but either the
C
2
operation or the
σ
v
operation alone would also interchange them; so, by looking only
at atom positions, it is impossible to tell which operation to chose. There is not enough
information in just the atom positions to tell the operations apart; some way is needed to
